Quantum Systems at The Brink

TL;DR

This paper introduces a higher-order correction method to the WKB approach for analyzing eigenfunctions of Schrödinger operators at the essential spectrum threshold, demonstrated through quantum particle examples.

Contribution

It develops a novel asymptotic analysis technique that extends WKB methods to threshold energies in quantum systems.

Findings

Effective in calculating eigenfunction behavior at spectrum thresholds

Applicable to quantum particles in complex potential wells

Enhances understanding of long-range potential effects

Abstract

We present a method to calculate the asymptotic behavior of eigenfunctions of Schr\"odinger operators that also works at the threshold of the essential spectrum. It can be viewed as a higher order correction to the well-known WKB method which does need a safety distance to the essential spectrum. We illustrate its usefulness on examples of quantum particles in a potential well with a long-range repulsive term outside the well.